Let $M$ be a $C^2$ compact Riemannian manifold with boundary.
Suppose $f \in L^2(M)$ is such that $0 \leq f \leq 1$. Is it possible to find $f_n \in C^\infty(M)$ such that $0 \leq f_n \leq 1$ for all $n$ and $f_n \to f$ in $L^2(M)$?
If $M$ were a bounded set this I could prove via convolution and mollification. Is it possible to do this on a manifold too? But I don't know if I can just patch it all together. Also, does my manifold need to be $C^\infty$ to make sense of the smooth functions on it?
Yes, this is possible. It suffices to say that $C^\infty(M)$ functions are dense in $L^2(M)$. This is e.g. exercise 10 in section 4.4 of Taylor's Partial Differential Equations 1, second edition (they prove it for Sobolev spaces $H^s(M)$, where $H^0(M)$ is the same as $L^2(M)$).